## Linear Operators: Spectral operators |

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Page 1529

Then in this angle any

the factor exp (§":*-*) is exponentially small (as |z| -- 00 in the angle) relative to

any

...

Then in this angle any

**solution**of Lj = 0 whose asymptotic expansion begins withthe factor exp (§":*-*) is exponentially small (as |z| -- 00 in the angle) relative to

any

**solution**whose asymptotic expansion begins with the factor exp (-;"z"-"). Thus...

Page 1553

Suppose that q is bounded below on the interval [0, oo), and let f be a bounded

point A belongs to the essential spectrum of r: (b) if all

Suppose that q is bounded below on the interval [0, oo), and let f be a bounded

**solution**of the equation (2–1)f = 0. Prove: (a) either f is square-integrable, or thepoint A belongs to the essential spectrum of r: (b) if all

**solutions**of the equation ...Page 1556

What is the relationship between 0(t) and the number of zeros of a

above equation? G14 Use the result of the preceding exercise to show that if the

operator r has two boundary values at infinity, then N(t) lim — = 00, t— oc t?

What is the relationship between 0(t) and the number of zeros of a

**solution**of theabove equation? G14 Use the result of the preceding exercise to show that if the

operator r has two boundary values at infinity, then N(t) lim — = 00, t— oc t?

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero